SYSTEM IDENTIFICATION OF CLASSIC HVDC SYSTEMS

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1 Vl.10(4) December 011 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS 113 SYSTEM IDENTIFICATION OF CLASSIC HVDC SYSTEMS L. Chetty* an N.M. Ijumba** *HVDC Centre, University f KwaZulu Natal,Private Bag X54001, 4001, Suth Africa chettyl@ukzn.ac.za ** HVDC Centre, University f KwaZulu Natal, Private Bag X54001, 4001, Suth Africa ijumban@ukzn.ac.za Abstract: Determining mels frm bservatins an stuying the mels prperties is essentially the functinality f science. Mels attempt t link bservatins int sme pattern. System ientificatin is the art f builing mathematical mels f ynamic systems base n bserve ata frm the systems. This paper presents a system ientificatin methlgy that can be utilize t erive the classic HVDC plant transfer functins. The mel evelpment an verificatin was perfrme using the PSCAD/EMTDC sftware. The calculate results illustrate excellent respnse matching with the system results. The erive HVDC plant transfer functins can be utilize t perfrm small signal stability stuies f HVDC-HVAC interactins an its use can als be extene t facilitate the analytical esign f HVDC cntrl systems. Key wrs: HVDC transmissin cntrl, melling, simulatin, uncertainty 1. INTRODUCTION Frm the early ays f HVDC system applicatins, the imprtance f mathematical melling f HVDC systems has been appreciate [1-10]. Mathematical mels f HVDC systems can be utilize t quantitatively analysis HVDC peratin issues which inclue secn harmnic instability an prblems relate t lw shrt circuit capacity f ac systems cnnecte t c systems [1,4,8]. A mathematical mel f an HVDC system can als be utilize fr the esign synthesis f HVDC cntrl systems [3]. There are essentially tw methlgies use t evelp mathematical mels f ynamic systems. One methlgy is t figuratively ivie the system int subsystems. The prperties f each subsystem are efine by the laws f nature an ther well-establishe relatinships [18]. These subsystems are mathematically cnsummate t frmulate a mel f the riginal system. Basic techniques f this methlgy invlve escribing the system s prcesses using blck iagrams. This methlgy is calle White Bx Melling [18]. The ther methlgy use t erive mathematical mels f a ynamic system is base n experimentatin [18]. Input an utput signals frm the riginal system are recre t infer a mathematical mel f the system. This methlgy is knwn as System Ientificatin [18]. each blck in the meshe system were erive using the state variable apprach. The transfer functins escribing the ac an c interactins were erive using escribing functin analysis. Perssn [3] calle these transfer functins cnversin functins. Figure 1: Blck iagram f HVDC transmissin system accring t [3] Base n the assumptin that the classic HVDC system is Base n the assumptin that the classic HVDC system is linear with regar t small variatins in the firing angle, linear with regar t small variatins in the firing angle, Freris et al. [4] evelpe a blck iagram, illustrate in Freris et al. [4] evelpe blck iagram, illustrate in Figure, t calculate the transfer functin f the rectifier Figure, t calculate the transfer functin f the rectifier current cntrl lp. Cntinuus wave mulatin an current cntrl lp. Cntinuus wave mulatin an Furier analysis were use t etermine the transfer Furier analysis were use t etermine the transfer functins f each blck in the meshe blck iagram. The functins f each blck in the meshe blck iagram. The cntinuus wave mulatin technique was use as a cntinuus wave mulatin technique was use as meth f evelping the escribing functins t accunt meth f evelping the escribing functins t accunt fr the ac/c interactins. fr the ac/c interactins.. MODELLING OF CLASSIC HVDC SYSTEMS: STATE OF THE ART Traitinally classic HVDC systems have been treate as linear time invariant systems [3-11]. Base n this premise, Perssn [3] evelpe a meshe blck iagram, illustrate in Figure 1, t calculate the current cntrl lp plant transfer functin. The transfer functins f Figure : Blck iagram f HVDC transmissin system Figure : Blck iagram f HVDC transmissin system accring t [4] accring t [4] Frm the linear time invariant system funatin, W et al. [5] perfrme Furier analysis n the c vltage an

2 114 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS Vl.10(4) December 011 ac current wavefrms f the cnverter. Frm these analyses, transfer functins were btaine fr the c vltage an ac currents with respect t the phase vltages an c currents. These transfer functins accmmate variatins in the firing angle an the cmmutatin peri. The subsequent transfer functins facilitate the preictins f vltage wavefrm istrtin n the c sie f the cnverter, an the preictin f current wavefrm istrtin n the ac sie f the cnverter. Using the transfer functins erive in [5], W et al. [6] evelpe an expressin fr the cnverter c sie frequency epenent impeance. This expressin was evelpe using the state-variable apprach. Using the state-variable apprach an the frequency epenent impeance f the cnverter, W et al [7] erive the transfer functin fr the current cntrl lp. Jvcic et al. [8], assume that classic HVDC systems are linear time invariant systems, therefre evelpe the plant transfer functin f the current cntrl lp using a state-variable apprach an the blck iagram illustrate in Figure 3. Figure 3: Blck iagram f HVDC transmissin system accring t [5] The state variables were chsen t be the instantaneus values f currents in the inuctrs an vltages acrss the capacitrs. In rer t represent the ac system ynamics tgether with the c system ynamics in the same frequency frame, the effect f the frequency cnversin thrugh the AC-DC cnverter was accmmate using Park s transfrmatin. The evelpe system mel was linearize arun the nrmal perating pint, an all states were represente as q cmpnents f the crrespning variables. The phase lcke scillatr was incrprate int the system mel. A review f the current state f the art f melling classic HVDC systems clearly inicates that the techniques utilize t evelp mathematical mels f classic HVDC systems have use the White Bx Melling methlgy. This methlgy requires accurate knwlege f the ac systems an the c systems an invlves cmplicate mathematics. In practice, it is nearly impssible t btain accurate knwlege f the ac systems cnnecte t classic HVDC systems. Als the limite time cnstraints impse n HVDC cntrl practitiners, the ac system uncertainties an the cmplicate mathematics have prevente the wiesprea practical use f the White Bx Melling methlgy t erive the plant transfer functins f classic HVDC systems. The bjective f this stuy was t utilize the System Ientificatin methlgy t erive mathematical mels f the classic HVDC systems. 3. SYSTEM IDENTIFICATION METHODOLOGY System ientificatin is the art f builing mathematical mels f ynamic systems base n bserve ata frm the systems [18]. A key cncept in utilizing the system ientificatin is the efinitin f the ynamic system upn which experimentatin can be cnucte. Manitba HVDC Research Centre cmmissine a stuy t examine the valiity f igitally efining the classic HVDC system [19-0]. T examine the valiity f igitally efining the classic HVDC system, the Nelsn River HVDC system was efine an simulate using the EMTDC prgram. EMTDC is a FORTRAN prgram an was use t represent an slve the linear an nn-linear ifferential equatins f electrmagnetic systems in the time main. A cmparisn was cnucte between the actual real-time system respnses an the igitally erive respnses. The results f the stuy illustrate that the igitally erive respnses crrelate excellently with the real system respnses. The stuy cnclue that the EMTDC prgram is a vali ptin fr igitally efining a classic HVDC system [19-0]. Therefre in this stuy, the classic HVDC system will be efine an experimente upn using the PSCAD/EMTDC prgram. PSCAD is a graphical user interface that easily assiste in efining the classic HVDC systems linear an nn-linear ifferential equatins in EMTDC. 3.1 Jacbian Linearizatin Cnsier a nn-linear system efine by the fllwing ifferential equatin: Where x t f ( x) g( x) u (1) y h(x) () x = the state variable vectr u = the input vectr y = the utput vectr

3 Vl.10(4) December 011 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS 115 The Jacbian linearizatin f the abve nn-linear system at a stable perating pint u, x, y is efine as x t y f ( x ) g ( x ) u ( xx ) g( x )( uu ) x x (3) h( x x ) ( x x y ) (4) Equatins (3) an (4) can be written in stanar linear state space representatin as [14]: x t Ax Bu (5) There are efinitive mes f peratin f the classic HVDC system, which are explicitly escribe as [17]: 1. Rectifier in Current Cntrl an the Inverter in Vltage Cntrl. Rectifier in Vltage Cntrl an the Inverter in Current Cntrl Therefre in the next sectins, methlgies utilize t erive the Rectifier Current Cntrl transfer functin an Inverter Current Cntrl transfer functin will be efine. 3. Rectifier Current Cntrl Plant Transfer Functin The classic HVDC system, shwn in Figure 4, was melle in PSCAD/EMTDC. y Cx (6) Where A, B, C = cnstant matrices The mel escribe by equatins (5) an (6) is a linear apprximatin f the riginal nn-linear system, escribe by equatins (1) an (), arun the stable perating pint u, x, y. This implies that riginal nn-linear system can be linearize if it can be cnstrue t functin arun a nrmal stable perating pint. The abve fact is explite in this stuy. The classic HVDC system was simulate in PSCAD/EMTDC t reach a nrmal steay-state perating pint. The classic HVDC system is cnsiere linearize arun the nrmal steay-state perating pint. Therefre a classic HVDC system can be cnsiere as linear time invariant system arun the stable perating pint. The impulse respnse f a linear time invariant system is etermine by first etermining the step respnse an then expliting the fact that the impulse respnse is btaine by ifferentiating the step respnse [16]. The Laplace transfrm f the impulse respnse is efine as the transfer functin f the linear time-invariant system [16]. The plant transfer functin can be explicitly btaine by etermining the rati f the Laplace transfrm f the step respnse t the Laplace transfrm f the step input [16]. This implies that the small signal plant transfer functin f a classic HVDC system can be btaine by etermining the rati f the Laplace transfrm f the small signal step respnse f the classic HVDC system t the Laplace transfrm f the step input f the rectifier firing angle r inverter firing angle. Figure 4: Simulate Fee-Frwar Cntrlle Classic HVDC System The system was simulate s that it reache steay-state. At which pint a snap sht f system variables was recre. The inverter firing angle was then kept cnstant. A fee-frwar step increase in the rectifier firing angle r, was execute an the c current respnse I was measure an is illustrate Figure 5. r Figure 5: Measure DC Current Respnse The measure current respnse was apprximate using the time main functin illustrate in equatin (7):

4 116 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS Vl.10(4) December 011 I r Where: T I 0 t T ( t) (7) at at I 1 e k I. e.sin( wt) t T = the time elay (sec) = the change in the c current (p.u.) k = cnstant ( 0 k 1); chsen t be 0.5 An: 1 a (8) T 1 I r Pcr( s). e T s r 3 s 3 3a 1s 3a a w k I r w s a a aw w k Ir aw s a s asa w (10) 3.3 Inverter Current Cntrl Plant Transfer Functin The classic HVDC system, shwn in Figure 4, was melle in PSCAD/EMTDC. The classic HVDC system was simulate s that it reache steay-state. At which pint a snap sht f system variables was recre. The rectifier firing angle was then kept cnstant. A feefrwar step ecrease in the inverter firing angle i was execute an the c current respnse I i was measure an is illustrate Figure 7. Where: w (9) T T 1 = the time (sec) it takes the ecaying wavefrm t 1 reach e f its final value. T = the peri (sec) f the superimpse as ac wavefrm. The functin escribe by equatin (7) is calle the HVDC Step Respnse (HSR) equatin an was simulate using MATLAB an the characteristic time main respnse is illustrate in Figure 6, tgether with the assciate errr when cmpare t the riginal signal. Figure 7: Measure DC Current Respnse The measure current respnse was apprximate using the HVDC Step Respnse (HSR) equatin as escribe in equatin (11): I i ( t) I 1 e at 0 k I. e at.sin( wt) t T (11) t T The HSR equatin was again simulate using MATLAB an characteristic time main respnse is illustrate in Figure 8, tgether with the assciate errr when cmpare t the riginal signal. Figure 6: Characterize DC Current Respnse Figure 6 clearly illustrates that the HSR equatin aequately apprximates the c current respnse t a step change in the rectifier s firing angle since the resultant errr es nt excee 1.5%. Equatin (7) was use t calculate the rectifier current cntrl plant transfer functin. The resultant rectifier current cntrl plant transfer functin is illustrate in the equatin (10):

Vl.10(4) December 011 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS 117 sectin, with the nly exceptin being that the effective shrt circuit ratis were varie.

es nt excee.0%. Equatin (9) was use t calculate the inverter current cntrl plant transfer functin. The resultant inverter current cntrl plant transfer functin is illustrate belw: I i Pci( s).

5 Vl.10(4) December 011 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS 117 sectin, with the nly exceptin being that the effective shrt circuit ratis were varie. Figure 8: Characterize DC Current Respnse Figure 8 clearly illustrates that the HSR equatin aequately apprximates the c respnse t a step change in the inverter s firing angle since the resultant errr es nt excee.0%. Equatin (9) was use t calculate the inverter current cntrl plant transfer functin. The resultant inverter current cntrl plant transfer functin is illustrate belw: I i Pci( s). e T s i 3 s 3 3a 1s 3a a w ki i ws a a aw w k Ii aw sa s asa w (1) 4. CLASSIC HVDC PLANT UNCERTAINTY In this stuy, the Thévenin s equivalent ac netwrk impeances were represente using a pure inuctance. This implies that the netwrk resistance is assume t be zer an the amping angle was taken as 90. Kunur [3] states that while lcal resistive las nt have a significant effect n the ESCR, these resistive las imprve the amping f the system thereby imprving the ynamic perfrmance f the cntrl system. By assuming the resistive element f the Thevenin s equivalent netwrk t be zer, the wrst case (frm a ynamic stability perspective) was simulate an analyse. Kunur [3] states that the ynamic perfrmance f a current cntrller is epenent n the strength f bth the rectifier an inverter ac systems. Therefre the variatins f the parameters f the rectifier current cntrl plant transfer functin an the inverter current cntrl plant transfer functin were calculate fr variatins in the rectifier cnverter statin s an the inverter cnverter statin s effective shrt circuit ratis. When the rectifier cnverter statin s ESCR varies frm.83 t 7.96 an the inverter cnverter statin s ESCR varies frm 3.93 t 7.96, the variatins in the rectifier current cntrl plant transfer functin parameters were etermine an the results are illustrate in Figure 9 an Table 1. The state f pwer systems change with suen isturbances in the pwer system. These suen isturbances will change the shrt circuit capacity f ac busbars in the pwer system. The factrs efining the quantitative change in shrt circuit capacity are lss f generatin, restratin f generatin, lss f transmissin, lss f eman an lss f reactive cmpensatin. Due t the iverse nature f the factrs affecting the quantitative change in shrt circuit capacity f an ac busbar implies that the shrt circuit capacity at a given HVDC cnverter ac busbar will vary within a range. Therefre cmbine with the varying amunt f c pwer that will be transmitte n the HVDC transmissin system, the effective shrt circuit rati (ESCR) fr a given HVDC cnverter statin will vary within a certain range. Table 1: Parametric Variatins f Rectifier Current Cntrl Plant Transfer Functin fr Varying ESCRs Due t the uncertain nature f the effective shrt circuit rati f rectifier an inverter cnverter statins, the plant transfer functins evelpe in the previus sectin will have a range f uncertainty. The bjective f this sectin f the paper will be t present the plant transfer functin parametric ranges fr varying shrt circuit ratis. The methlgies use t calculate the parametric variatins in the plant transfer functins were exactly the same as the methlgies presente in the previus

118 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS Vl.

Parametric Variatins Figure 10: Inverter Current Cntrl Plant Transfer Functin

8 an the inverter cnverter statin s ESCR varies frm 3.94 t 11.

parameters were etermine an the results are illustrate in Figure 10 an Table.

the ac system s effective shrt circuit rati is knwn, the range f parametric

Table : Parametric Variatins f Inverter Current Cntrl Plant Transfer Functin

DISCUSSSION The current state f the art f evelping mathematical mels f

This paper presents a System Ientificatin methlgy t erive the plant transfer

The classic HVDC system linear an nn-linear ifferential equatins were melle

Fr small changes in the rectifier s firing angle, the rectifier s current

6 118 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS Vl.10(4) December 011 Figure 9: Rectifier Current Cntrl Plant Transfer Functin Parametric Variatins Figure 10: Inverter Current Cntrl Plant Transfer Functin Parametric Variatins When the rectifier cnverter statin s ESCR varies frm 4 t 8 an the inverter cnverter statin s ESCR varies frm 3.94 t 11.8, the variatins in the inverter current cntrl plant transfer functin parameters were etermine an the results are illustrate in Figure 10 an Table. Frm the abve presente results, it can easily be ientifie that if the range f the ac system s effective shrt circuit rati is knwn, the range f parametric uncertainty f the classic HVDC plant transfer functins can be btaine. Table : Parametric Variatins f Inverter Current Cntrl Plant Transfer Functin fr Varying ESCRs 5. DISCUSSSION The current state f the art f evelping mathematical mels f classic HVDC systems has been t utilize the White Bx Melling methlgy. This paper presents a System Ientificatin methlgy t erive the plant transfer functins f classic HVDC systems. The classic HVDC system linear an nn-linear ifferential equatins were melle in PSCAD/EMTDC. Fr small changes in the rectifier s firing angle, the rectifier s current cntrl lp can be linearize arun a stable (r equilibrium) peratin pint. This is efine as the Jacbian Linearizatin f the riginal nn-linear current cntrl lp. PSCAD/EMTDC was use t btain the c current step respnse f a classic HVDC system. The measure current respnse was apprximate using the time main functin efine as HVDC Step

7 Vl.10(4) December 011 SOUTH AFRICAN INSTITUTE OF ELECTRICAL ENGINEERS 119 Respnse (HSR) equatin. The results valiate that HSR equatin aequately apprximate the c current respnse t a step change in the rectifier s an inverter s firing angle since the resultant errr was very minimal. The Laplace transfrm f the characterize c current respnse an the firing angle step input were etermine an subsequent rati f these Laplace transfrmatins pruce the HVDC Rectifier an Inverter Current Cntrl Plant transfer functins. Due t the uncertain nature f the state f pwer systems, the parameters f the plant transfer functins that efine the classic HVDC systems vary. The range f plant transfer functin parametric variatins was etermine as a functin f ac systems effective shrt circuit rati. Therefre if the range f the ac system s effective shrt circuit rati is knwn, the range f parametric uncertainty f the classic HVDC plant transfer functins can be btaine. 6. REFERENCES [1] K. Erikssn, G. Liss, E.V. Perssn: "Stability Analysis f the HVDC Cntrl System Transmissin Using Theretically Calculate Nyquist Diagrams", IEEE Transactins n Pwer Apparatus an Systems, Vl. PAS-89, N.5/6, pp , June [] E. Rumpf an S. Ranae: "Cmparisn f Suitable Cntrl Systems fr HVDC Statins cnnecte t Weak AC Systems Part1: New Cntrl Systems", IEEE Transactins n Pwer Apparatus an Systems, Vl. PAS-91, N., pp , March 197. [3] E.V. Perssn: "Calculatin f transfer functins in gri-cntrlle cnverter systems", IEE Prceeings Vl. 117, N.5, pp , May [4] J.P. Sucena-Paiva an L.L. Freris: "Stability f a c transmissin link between weak ac systems", IEE Prceeings, Vl. 11, N. 6, pp , June [5] A.R. W an J. Arrillaga: "HVDC cnverter wavefrm istrtin: a frequency main analysis", IEE Prceeings, Vl.14, N.1, pp , January [6] A.R. W an J. Arrillaga: "Cmpsite resnance; a circuit apprach t the wavefrm istrtin ynamics f an HVDC cnverter", IEEE Transactins n Pwer Delivery, Vl.10, N.4, pp , Octber [7] A.R. W an J. Arrillaga: "The frequency epenent impeance f an HVDC cnverter", IEEE Transactins n Pwer Delivery, Vl.10, N.3, pp , July [8] D. Jvcic, N. Pahalawaththa, M. Zavahir: "Analytical melling f HVDC-HVAC systems", IEEE Transactins n Pwer Delivery, Vl. 14, N., pp , April [9] D. Jvcic, N. Pahalawaththa, M. Zavahir: "Stability analysis f HVDC cntrl lps", IEE Prceeings - Generatin, Transmissin an Distributin, Vl. 146, N., pp , March [10] M. Aten, K.M. Abbtt, N. Jenkins: "Develpments in melling an analysis f HVDC cntrl systems", Prceeings f Eurpean Pwer Electrnics an Applicatins Cnference, Graz, 001. [11] A.H.M.A. Rahim an E.P. Nwicki: "A Rbust Damping Cntrller fr an HV-ACDC System using a Lp-Shaping Prceure", Jurnal f Electrical Engineering, Vl. 56, N.1-, pp. 15-0, 005. [1] J. Karlssn: Simplifie Cntrl Mel fr HVDC Classic, MSc. Thesis, Ryal Institute f Technlgy, Stckhlm, 006 [13] Manitba HVDC Research Centre Inc.: PSCAD/EMTDC 4. User s Manual, 005 [14] Packar, Pla, Hrwitz: "Dynamic Systems an Feeback" Class Ntes fr ME13, University f Califrnia, 00 [15] IEEE/CIGRE Jint Task Frce: "Definitin an Classificatin n Pwer System Stability" IEEE Transactins n Pwer Systems, Vl. 19, N., pp , May 004. [16] S. Haykin an B. van Veen; Signals an Systems, Jhn Wiley & Sns, 001 [17] P. Kunur: Pwer System Stability an Cntrl, ISBN X, McGraw-Hill, 1994 [18] L. Ljung: System Ientificatin - Thery fr the User. Prentice-Hall, USA, n eitin, [19] D.A. Wfr: "Valiatin f Digital Simulatin f DC Links" IEEE Transactins n Pwer Apparatus an Systems, PAS-104, N. 9, pp , September [0] T. In, R.M. Mathur, M.R. Iravani an S. Sasaki: "Valiatin f Digital Simulatin f DC Links - Part II", IEEE Transactins n Pwer Apparatus an Systems, Vl. PAS-104, N. 9, pp , September 1985.

Design and Simulation of Dc-Dc Voltage Converters Using Matlab/Simulink

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